### Video Transcript

Determine the integral of the cos
of seven ๐ฅ minus seven with respect to ๐ฅ.

The question is asking us to
evaluate the integral of a composite function. Itโs the integral of the
composition of a trigonometric function and a linear function. And thereโs a lot of different ways
we could think about evaluating this integral. For example, we might want to try
using our multiple angle formulas to rewrite our integrand. However, this is also the integral
of a composite function. So we should also check if thereโs
a suitable ๐ข substitution we can use to make this integral easier. And in this case, this is true. Weโll set ๐ข to be our inner
function. Thatโs the linear function seven ๐ฅ
minus seven. Then, we can differentiate ๐ข with
respect to ๐ฅ. Since this is a linear function, we
just get the coefficient of ๐ฅ which is seven.

At this point, we need to be a
little bit careful. Normally, we would check to see if
d๐ข by d๐ฅ appears in our integrand so we could use this to make or integral
simpler. However, on first glance, this
doesnโt seem to be the case. However, if we multiply our
integrand by seven and then multiply our entire integral by one over seven, we donโt
change the value of our integral. But now we have d๐ข by d๐ฅ
appearing in our integrand. And in doing this, weโve rewritten
our integral to be of the form one-seventh times the integral of ๐ prime of ๐ฅ
times ๐ of ๐ of ๐ฅ with respect to ๐ฅ. We can use the reverse chain
rule.

So letโs try and evaluate our
integral by using the substitution ๐ข is equal to seven ๐ฅ minus seven. Weโve already shown d๐ข by d๐ฅ is
equal to seven. And remember, d๐ข by d๐ฅ is not a
fraction. However, when weโre integrating by
substitution, we can treat it a little bit like a fraction. This gives us the equivalent
statement in terms of differentials d๐ข is equal to seven d๐ฅ. Weโre now ready to evaluate our
integral by using substitution. First, the integral of the cos of
seven ๐ฅ minus seven with respect to ๐ฅ is equal to one-seventh times the integral
of seven times the cos of seven ๐ฅ minus seven with respect to ๐ฅ.

Next, we set ๐ข to be our inner
function seven ๐ฅ minus seven. We then show that d๐ข by d๐ฅ is
equal to seven. And this is equivalent to saying
d๐ข is equal to seven d๐ฅ. In other words, in our integration
by substitution, we can replace seven d๐ฅ with d๐ข. So using integration by
substitution, we get one-seventh multiplied by the integral of the cos of ๐ข with
respect to ๐ข. And we know how to evaluate this
integral. The integral of the cos of ๐ with
respect to ๐ is equal to the sin of ๐ plus the constant of integration ๐. So applying this rule with ๐ข
instead of ๐, we get one-seventh times the sin of ๐ข plus the constant of
integration weโll call ๐ one.

Distributing one-seventh over our
parentheses, we get one-seventh times the sin of ๐ข plus ๐ one over seven. And weโll simplify our answer. First, ๐ one is a constant. So ๐ one divided by seven is also
a constant. So weโll just replace this with a
new constant weโll call ๐. Next, remember, our original
integral was in terms of ๐ฅ. So we want to give our answer in
terms of ๐ฅ. Weโll do this by reusing our
substitution ๐ข is equal to seven ๐ฅ minus seven. This gives us one-seventh times the
sin of seven ๐ฅ minus seven plus ๐. And this is our final answer.

Therefore, by using integration by
substitution, weโve shown the integral of the cos of seven ๐ฅ minus seven with
respect to ๐ฅ is equal to one-seventh times the sin of seven ๐ฅ minus seven plus our
constant of integration ๐.